Theorem. Let the function $f(z) = u(x, y) + iv(x, y)$ be analytic in a domain $D$, and Let the families of level curves be $u(x, y) = c_1$ and $v(x, y) = c_2$, where $c_1$ and $c_2$ are arbitrary real constants. Then these families are orthogonal.
Regarding the mentioned Theorem the two following examples are confusing:
Consider $f(z)=z^2$. I plot the $x^2-y^2=a$ and $2xy=b$ for different values of a and b and they are orthogonal and I tried for a=b=0 but it fails to be orthogonal! Does it contradict with the above theorem?
And, consider $f(z)=\dfrac{1}{z}$. I plot the $x/(x^2+y^2)=a$ and $y/(x^2+y^2)=b$ for different values of a and b and they are orthogonal and I tried for a=b=0 and still it doesn't fail to be orthogonal but $f(z)=\dfrac{1}{z}$ is not analytic for $z=0$! Does it contradict with the above theorem?
Best Answer
The theorem is only valid for those points at which $f'$ is not $0$. The explains your first question.
Concerning the second one, the function isn't even defined at $0$.